International audienceThe scenario of transition to chaos for a sphere falling or ascending under the action of gravity in a Newtonian fluid is investigated by numerical simulation. The mathematical formulation is parameterized using two non-dimensional parameters: the solid/fluid density ratio and the generalized Galileo number expressing the ratio between the gravity–buoyancy and viscosity effects. The study is carried out fully in this two-parameter space. The results show that for all density ratios the vertical fall or ascension becomes unstable via a regular axisymmetry breaking bifurcation. This bifurcation sets in slightly earlier for light spheres than for dense ones. A steady oblique fall or ascension follows before losing stability and giving way to an oscillating oblique movement. The secondary Hopf bifurcation is shown not to correspond to that of a fixed sphere wake for density ratios lower than 2.5, for which the oscillations have a significantly lower frequency. Trajectories of falling spheres become chaotic directly from the oblique oscillating regime. Ascending spheres present a specific behaviour before reaching a chaotic regime. The periodically oscillating oblique regime undergoes a subharmonic transition yielding a low-frequency oscillating ascension which is vertical in the mean (zigzagging regime). In all these stages of transition, the trajectories are planar with a plane selected randomly during the axisymmetry breaking. The chaotic regime appears to result from an interplay of a regular and of an additional Hopf bifurcation and the onset of the chaotic regime is accompanied by the loss of the remaining planar symmetry. The asymptotic chaotic states present an intermittent character, the relaminarization phases letting the subcritical plane and periodic trajectories reappear

The first Hopf bifurcation of the infinite cylinder wake is analysed theoretically and by direct simulation. It is shown that a decomposition into a series of harmonics is a convenient theoretical and practical tool for this investigation. Two basic properties of the instability allowing the use and truncation of the series of harmonics are identified: the lock-in of frequencies in the flow and separation of the rapid timescale of the periodicity from the slow timescale of the non-periodic behaviour. The Landau model is investigated under weak assumptions allowing strong nonlinearities and transition to saturation of amplitudes. It is found to be rather well satisfied locally at a fixed position of the flow until saturation. It is shown, however, that no truncated expansion into a series of powers of amplitude can account correctly for this fact. The validity of the local Landau model is found to be related to the variation of the form of the unstable mode substantially slower than its amplification. Physically relevant characteristics of the Hopf bifurcation under the assumption of separation of three timescales – those of the periodicity, amplification and deformation of the mode – are suggested.

The primary and secondary instabilities of the sphere wake are investigated from the viewpoint of nonlinear dynamical systems theory. For the primary bifurcation, a theory of axisymmetry breaking by a regular bifurcation is given. The azimuthal spectral modes are shown to coincide with nonlinear modes of the instability, which provides a good reason for using the azimuthal expansion as an optimal spectral method. Thorough numerical testing of the implemented spectral–spectral-element discretization allows corroboration of existing data concerning the primary and secondary thresholds and gives their error estimates. The ideal axisymmetry of the numerical method makes it possible to confirm the theoretical conclusion concerning the arbitrariness of selection of the symmetry plane that arises. Investigation of computed azimuthal modes yields a simple explanation of the origin of the so-called bifid wake and shows at each Reynolds number the coexistence of a simple wake and a bifid wake zone of the steady non-axisymmetric regime. At the onset of the secondary instability, basic linear and nonlinear characteristics including the normalized Landau constant are given. The periodic regime is described as a limit cycle and the power of the time Fourier expansion is illustrated by reproducing experimental r.m.s. fluctuation charts of the streamwise velocity with only the fundamental and second harmonic modes. Each time–azimuthal mode is shown to behave like a propagating wave having a specific spatial signature. Their asymptotic, far-wake, phase velocities are the same but the waves keep a fingerprint of their passing through the near-wake region. The non-dimensionalized asymptotic phase velocity is close to that of an infinite cylinder wake. A reduced-accuracy discretization is shown to allow qualitatively satisfactory unsteady simulations at extremely low cost.

Detailed data describing the motion of a rigid sphere settling in unperturbed fluid is generated by means of highly-accurate spectral/spectral-element simulations with the purpose of serving as a future benchmark case. A single solid-to-fluid density ratio of 1.5 is chosen, while the value of the Galileo number is varied from 144 to 250 such as to cover the four basic regimes of particle motion (steady vertical, steady oblique, oscillating oblique, chaotic). This corresponds to a range of the particle Reynolds number from 185 to 365. In addition to the particle velocity data, extracts of the fluid velocity field are provided, as well as the pressure distribution on the sphere's surface. Furthermore, the same solid-fluid system is simulated with a particular non-boundary-conforming approach, i.e. the immersed boundary method proposed by Uhlmann (2005a), using various spatial resolutions. It is shown that the current benchmark case allows to adjust the resolution requirements for a given error tolerance in each flow regime.